Deformations of nilpotent groups and homotopy symmetric $C^{\ast }$-algebras

Abstract

The homotopy symmetric $C^{\ast }$-algebras are those separable $C^{\ast }$-algebras for which one can unsuspend in E-theory. We find a new simple condition that characterizes homotopy symmetric nuclear $C^{\ast }$-algebras and use it to show that the property of being homotopy symmetric passes to nuclear $C^{\ast }$-subalgebras and it has a number of other significant permanence properties. As an application, we show that if $I(G)$ is the kernel of the trivial representation $\iota :C^{\ast }(G)\to \mathbb{C}$ for a countable discrete torsion free nilpotent group $G$, then $I(G)$ is homotopy symmetric and hence the Kasparov group $KK(I(G),B)$ can be realized as the homotopy classes of asymptotic morphisms $[[I(G),B\otimes \mathcal{K}]]$ for any separable $C^{\ast }$-algebra $B$.

13p

References in corpus (2)

• K-Theory and Pseudospectra for Topological Insulators
• Representation spaces for central extensions and almost commuting unitary matrices

Cited by in corpus (3)

• Connective C*-algebras
• Deformations of Wreath Products
• Inductive limits of semiprojective C*-algebras